Determining the strength of undetectable particle-hole configurations by complete spectroscopy of negative parity states in ${}^{208}\mathrm{Pb}$

In the doubly magic nucleus ${}^{208}\mathrm{Pb}$, many states contain fractions of particle-hole configurations whose strength can be determined from experiment. However, some configurations are not excited in a directly detectable way. Their strengths can be determined by observing an ensemble of states which consists entirely of an equivalent set of configurations with a given spin and parity among which only one or two configurations are not detected. Examples for spins $2^{-}$–$5^{-}$ are evaluated. Excitation energies of states in ${}^{208}\mathrm{Pb}$ are determined with a precision down to 100 eV by experiments on the ${}^{208}\mathrm{Pb}$$\left(p,p^{\prime }\right)$ and ${}^{207}\mathrm{Pb}$$\left(d,p\right)$ reactions with the Q3D magnetic spectrograph (Maier-Leibnitz-Laboratorium, Garching, Germany). Six doublets with distances between the states of less than 2 keV are resolved. 72 negative parity states below $E_x=6.1$ MeV are identified. They correspond to 70 states predicted by the schematic shell model without residual interaction below $E_x=6361$ keV. The $1^{-}$ and $3^{-}$ yrast states appear in addition. Six new spins are assigned to negative parity states, three new spins to positive parity states, and two spins suggested by the Nuclear Data Sheets are verified. The state at $E_x=4953$ keV is identified as the 3${}^{+}$ member of the configuration $g_{9/2}$$i_{13/2}$. Among about hundred states, the configuration mixing for unnatural parity is shown to be less than for natural parity.

Figure 1

Transformation matrix $\left|\right|c\left|\right|$ [Eqs. (1), (4), and (5)] for states in ${}^{208}\mathrm{Pb}$ with spin $2^{-}$. For each configuration, the order number $m$, the SSM energy, and the orbitals $LJ$ and $lj$ of the particle and the hole are given along the abscissa. The size of the rectangle shows the strength ${\left[c_{LJ,lj}^{{\tilde{E}}_x{I}^{\pi }}\right]}^2$; open rectangles denote the undetectable proton configuration $f_{7/2}$$d_{3/2}$.

Figure 2

Calculated angular distributions of pure particle-hole configurations $LJlj$ with spins $|J-j|\le I\le J+j$ for $lj=f_{5/2}$ and $LJ=d_{5/2}$, $g_{9/2}$, $j_{15/2}$. Angular distributions of $g_{9/2}$$f_{7/2}$ for the lowest and the highest spin are shown too. The different line styles are explained in the upper right frame.

Figure 3

Excerpts of spectra from ${}^{208}\mathrm{Pb}$$\left(p,p^{\prime }\right)$ via IAR in ${}^{209}\mathrm{Bi}$ for $5.18<E_x<5.51$ MeV. For details see Sec. 3b.

Figure 4

Excerpts of spectra from ${}^{208}\mathrm{Pb}$$\left(p,p^{\prime }\right)$ via IAR in ${}^{209}\mathrm{Bi}$ for $5.46<E_x<5.57$ MeV. For details see Sec. 3b.

Figure 5

Excerpts of spectra for $3.91<E_x<4.24$ MeV (a) on the $g_{9/2}$ IAR, (b) on the $i_{11/2}$ IAR, (c) near the $d_{5/2}$ IAR ($E^{\mathrm{res}}=16.496$ MeV [5]), and (d) on the $s_{1/2}$ IAR. The states with dominant $g_{9/2}$$f_{5/2}$ components are marked in frame (a), the states with dominant $i_{11/2}$$p_{1/2}$ components in frame (b), and the $4^{-}$ state with a dominant proton configuration $h_{9/2}$$s_{1/2}$ component and the 4086 $2^{+}$ state in frame (d). (The 3708 $5^{-}$ and 3961 $5^{-}$ states contain the major $h_{9/2}$$s_{1/2}$ $5^{-}$ strength.) Satellites from the electron knockout reaction [13] are especially observed for the 4085 2${}^{+}$ state. For more details see Sec. 3b.

Figure 6

Continuation of Fig. 5 for $4.24<E_x<4.77$ MeV. [In frame (d), the end of the spectrum is distorted.] The states with dominant $g_{9/2}$$p_{3/2}$ and $i_{11/2}$$f_{5/2}$ components are marked in frames (a) and (b), and the states with the dominant proton configuration $h_{9/2}$$d_{3/2}$ in frame (d); for the $5^{-}$ member see Fig. 5. The 4324 $4^{+}$, 4424 $6^{+}$, 4611 $8^{+}$ states are marked in frame (c). Satellites from the electron knockout reaction [13] are especially observed for the 4698 3${}^{-}$ state. For more details see Sec. 3b.

Figure 7

Excerpts of spectra from ${}^{208}\mathrm{Pb}$$\left(p,p^{\prime }\right)$ near the $g_{7/2}$, $d_{3/2}$ IARs in ${}^{209}\mathrm{Bi}$ ($E^{\mathrm{res}}=17.430$, 17.476 MeV, respectively [5]) taken consecutively (a) for $4.55<E_x<4.80$ MeV and (b) for $4.80<E_x<5.03$ MeV; in Fig. 8 for $5.02<E_x<5.46$ MeV, in Fig. 9 for $5.46<E_x<5.85$ MeV, and in Fig. 10 for $5.85<E_x<6.29$ MeV. For details see Sec. 3b.

Figure 8

Continuation of Fig. 7, (a) for $5.02<E_x<5.19$ MeV and (b) for $5.19<E_x<5.46$ MeV.

Figure 9

Continuation of Figs. 7–8, (a) for $5.46<E_x<5.63$ MeV and (b) for $5.63<E_x<5.86$ MeV. The doublet 5812 $2_7^{-}$, 5813 $3_{13}^{-}$ is fitted by assuming a distance of 0.5 keV [20].

Figure 10

Continuation of Figs. 7–9, (a) for $5.86<E_x<6.05$ MeV and (b) for $6.05<E_x<6.29$ MeV. Except for the $1_6^{-}$, $2_9^{-}$, $3_{17}^{-}$ states, no negative parity state in the region $6.02<E_x<6.21$ MeV is known. The gap corresponds to the predicted gap $5922\le E_x^{\mathrm{SSM}}\le 6361$ keV for spins 4${}^{-}$ - 7${}^{-}$ [13].